For which value of k will (k+3)x² + 2kx + (k−1) = 0 have equal roots?
Answer and explanation
Correct answer: k = 3/2
The governing concept is the discriminant criterion for equal roots. For a quadratic equation ax² + bx + c = 0 to have two real and equal roots, its discriminant D = b² − 4ac must be zero. Here a = k + 3, b = 2k, and c = k − 1. Therefore, D = (2k)² − 4(k+3)(k−1) = 4k² − 4(k² + 2k − 3) = 12 − 8k. Setting D equal to zero gives 12 − 8k = 0, so 8k = 12 and k = 3/2. At this value, a = 9/2, which is nonzero, so the equation remains genuinely quadratic. Hence option A is correct. The other choices do not make the discriminant zero; k = −3 would even remove the x² term.
Frequently asked questions
What is the correct answer to this question?
k = 3/2
Why is this the correct answer?
The governing concept is the discriminant criterion for equal roots. For a quadratic equation ax² + bx + c = 0 to have two real and equal roots, its discriminant D = b² − 4ac must be zero. Here a = k + 3, b = 2k, and c = k − 1. Therefore, D = (2k)² − 4(k+3)(k−1) = 4k² − 4(k² + 2k − 3) = 12 − 8k. Setting D equal to zero gives 12 − 8k = 0, so 8k = 12 and k = 3/2. At this value, a = 9/2, which is nonzero, so the equation remains genuinely quadratic. Hence option A is correct. The other choices do not make the discriminant zero; k = −3 would even remove the x² term.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Nature of Roots.
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